fundamentals of drug delivery€¦ · based drugs development of combinatorial chemistry & hts...

1

FUNDAMENTALS OF DRUGDELIVERY

Rebecca A. Bader

Syracuse Biomaterials Institute, Syracuse University, Syracuse, NY, USA

1.1 INTRODUCTION: HISTORY AND FUTURE OF DRUG DELIVERY

As depicted in Fig. 1.1, as drug discovery has evolved, the need for innovate meth-ods to effectively deliver therapeutics has risen. In the early 1900s, there began ashift away from the traditional herbal remedies characteristic of the “age of botani-cals” toward a more modern approach based on developments in synthetic chemistry[1, 2]. Through the 1940s, drug discovery needs were directed by the needs of themilitary, that is, antibiotics were developed and produced to treat injured soldiers[3]. As more pharmaceuticals were rapidly identified by biologists and chemists alike,people became more cognizant of the impact therapeutics could have on everyday life.During the late 1940s to the early 1950s, drugs were, for the first time, formulatedinto microcapsules to simplify administration and to facilitate a sustained, controlledtherapeutic effect [4]. For example, Spansules®, microcapsules containing drug pel-lets surrounded by coatings of variable thickness to prolong release, were developedby Smith Kline and French Laboratories and rapidly approved for use [5]. Many ofthese early microencapsulation techniques, particularly the Wurster process, wherebydrug cores are spray coated with a polymer shell, are still in use today [6, 7].

Engineering Polymer Systems for Improved Drug Delivery, First Edition.Edited by Rebecca A. Bader and David A. Putnam.© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

COPYRIG

HTED M

ATERIAL

4 FUNDAMENTALS OF DRUG DELIVERY

Conventionaldosage forms

The wurster process:microencapsulation of

drug particles

Introduction of polymer-drug conjugates

("nanotherapeutics")

Drug-elutingcoronary stents

approved

<1900 1930s 1940s 1950s 1960s 1970s 1980s 1990s >2000

<1900 1930s 1940s 1950s 1960s 1970s 1980s 1990s >2000

"Age ofchemical synthetic

discovery"

Emergence ofcomputer technology

"Age of biotechnology"

"Antibiotic era""Age ofbotanicals"

Approval offirst monoclonal

antibody

Approval of thespansule

Introduction ofdrug-loaded depots

and liposomes

Development of PEGylated liposomes

and micelles

Developmentof protein-

based drugs

Development ofcombinatorial

chemistry & HTS

Delivery systems thatsimulate physiologicalpatternsOral delivery ofpeptides/proteinsGene therapy

(a)

(b)

Figure 1.1. Drug delivery (a) and drug discovery (b) have followed similar trajectories with

the need for drug delivery rising with the identification of new therapeutic compounds.

Although a number of advanced methods for controlled and/or targeted drug deliv-ery were proposed in the 1960s, building on the conventional drug delivery methodof microencapsulation, these techniques were not fully implemented until the 1970s[8, 9]. During this decade, biotechnology and molecular biology began to play a sig-nificant role in the drug discovery process, culminating in an increased understandingof the etiology of numerous diseases and the development of protein-based therapeu-tics. Likewise, computer screening, predictive software, combinatorial chemistry, andhigh throughput screening significantly accelerated the rate at which lead compoundsfor new therapeutic compounds could be identified [1, 4]. As is discussed further inChapter 2, drug carrier systems, such as implants, coatings, micelles, liposomes, andpolymer conjugates, were proposed to address the growing need to deliver the newlyidentified therapeutic compounds with maximum efficacy and minimal risk of negativeside effects [8, 9] (Fig. 1.2).

In sum, over time, as technology has advanced for drug discovery, there has beena paradigm shift in drug delivery from simplifying the administration of old drugsto creating systems that can make new drugs work. This is particularly true as wecontinue to identify and develop therapeutics based on proteins and nucleic acids thatare difficult to administer in a patient-friendly manner and/or with the necessary site-specificity to reverse adverse consequences. However, as drug delivery technologyhas advanced for new drugs, many of the old drugs have likewise benefited throughincreased predictability of pharmacokinetic/pharmacodynamic profiles, decreased sideeffects, and enhanced efficacy. This text is intended to explain how these advanceddrug delivery techniques, particularly those related to the application of polymers, have

TERMINOLOGY 5

METABOLISMMetabolite formation

EXCRETION EXCRETION

ABSORPTION

PLASMA

Cp (bound) Cp(unbound) Ct (bound) Ct(unbound)

DISTRIBUTION TISSUE

Figure 1.2. The temporal and spatial distribution of drugs is impacted by absorption, distri-

bution, metabolism, and excretion (ADME).

improved the efficacy of old and new drugs alike. Chapter 1 serves as the foundationfor all subsequent chapters, defining the necessary terminology related to drug deliveryand pharmaceutics.

1.2 TERMINOLOGY

1.2.1 Pharmacology

Pharmacology, the science of drugs, is composed of two primary branches, pharma-codynamics and pharmacokinetic. In broad terms, pharmacokinetics refers to what thebody does to the drug whereas pharmacodynamics describes what the drug does to thebody. In the subsequent sections, a brief overview of these two branches of study aregiven in order to highlight some of the basic pharmacological terminology frequentlyencountered in both drug discovery and delivery

1.2.1.1 Pharmacokinetics. Pharmacokinetics tracks the time course of drugsand drug delivery systems through the body. The processes that impact the tempo-ral and spatial distribution of drugs are absorption, distribution, metabolism, excre-tion (ADME). Following administration, the drugs are absorbed by the bloodstream,


TABLE 1.1. Pharmacokinetic Parameters

Process Parameter Definition

Absorption Absorption rate constant (ka) First-order rate constant forabsorption

Bioavailability (F ) The extent of drug absorptionDistribution Plasma drug concentration (Cp) The concentration of drug in the

plasmaVolume of distribution (Vd) The mass amount of drug given

(dose) divided by the plasmaconcentration (Cp). Vd is anapparent volume with nodirect physiological relevance

Unbound fraction The fraction of drug not boundto protein, that is,pharmaceutically active

Elimination(metabolism andexcretion)

Metabolism rate constant (km) First-order rate constant forelimination by metabolism

Excretion rate constant (kex) First-order rate constants forelimination by excretion

Elimination rate constant (ke) ke = kex + kmExtrarenal (metabolic) clearance The volume of plasma cleared

of drug per unit time bymetabolism

Renal clearance The volume of plasma clearedof drug per unit time bymetabolism

Total clearance Total clearance = renalclearance + extrarenalClearance

Half-life (t1/2) The time necessary for theplasma drug concentration tobe reduced 50%

distributed to tissues and organs throughout the body, and eventually eliminated bymetabolism or excretion. Although a summary of these processes with associatedparameters is provided in Table 1.1, each of these terms are described in further detailin Section 1.3 [10, 11].

1.2.1.2 Pharmacodynamics. Because pharmacodynamics broadly refers towhat the drug does to the body, pharmacodynamics measurements involve lookingat toxicity, as well as therapeutic efficacy. These measurements frequently involveexamining dose–response curves to determine the optimal range over which drugscan be administered with maximum therapeutic impact and minimal negative sideeffects. Pharmacodynamics also involves examining the mechanism by which drugsact, that is, drug–receptor interactions. Typically, these studies are used to identify

TERMINOLOGY 7

the amount of drug necessary to reduce interactions of endogenous agonists with thereceptor [12]. These concepts related to pharmacodynamics will be explored in greaterdetail in Section 1.4.

1.2.2 Routes of Administration

The route by which drugs are administered can have a profound impact on the phar-macokinetic properties given in Table 1.1. One of the goals of drug delivery is tofacilitate administration by routes that normally have an adverse impact on the asso-ciated therapeutic pharmacokinetic properties. For example, as is discussed further inChapter 2, effective oral administration of numerous drugs is not feasible because ofpoor uptake through the mucosal epithelial barrier of the intestine and a low resultantbioavailability. Furthermore, orally administered drugs are subject to what is referredto as the first pass effect, whereby the bioavailability is reduced by metabolism withinthe liver and/or gut wall. Carrier systems have been designed to (i) increase intercel-lular transport by disrupting the epithelial barrier, (ii) facilitate intracellular transportthrough targeting of the absorptive epithelial cells, and/or (iii) reduce the destructionof drugs by liver enzymes [13–16].

The most explored routes of drug administration are summarized in Table 1.2.Although 90% of drugs are administered orally due to convenience and high patientcompliance, oral drug delivery is associated with low and/or variable bioavailabilityas a result of the harsh environment of the gastrointestinal tract and the impermeablenature of the mucosal epithelial barrier. In contrast, parenteral forms of adminis-tration (intravenous, subcutaneous, and intramuscular) yield rapid effects and highbioavailability (100% for intravenous); however, patient compliance is extremelylow as a result of the discomfort because of the injection. Transdermal delivery is

TABLE 1.2. Routes of Administration for Drug Delivery

Route ofAdministration

Advantages Limitations

Parenteral Immediate effectsReproducibleHigh bioavailability

Low patient complianceOften requires a clinician

Oral ConvenientHigh patient compliance

Highly variableHarsh environmental conditionsLow absorption of many drugs

Transdermal Continuous delivery Limited to lipophilic drugsPulmonary High absorptive surface area

Rapid absorption of small moleculedrugs

The morphology of the lung tissuemakes systemic delivery difficult

Limited absorption ofmacromolecules

Nasal Rapid absorption of lipophilic drugsHigh bioavailability of lipophilic

drugs

Limited absorption of polarmolecules


a favorable route of administration because of high patient acceptability and readyaccess to the site of absorption; however, this method has historically been limitedto small, lipophilic drugs that can passively diffuse through the skin barrier [17, 18].New techniques are currently being developed to extend transdermal delivery to polarand/or macromolecular compounds. For example, ultrasound and iontophoresis pro-vide a driving force for the passage of small, charged drugs, while electroporation andmicroneedles disrupt the outermost layer of the skin for delivery of macromolecules,particularly peptides and proteins [19]. Nasal and pulmonary drug deliveries are alsoattractive routes of administration because of the high potential surface area availablefor drug absorption; however, as with transdermal delivery, the nature of the epithelialbarriers in both regions limits this to lipophilic compounds [17, 18].

1.2.3 Drug Delivery

1.2.3.1 Controlled Release. Controlled drug delivery systems, also referredto as prolonged and sustained release systems, aim to minimize dosing frequency bymaintaining the local and/or systemic concentrations of drugs for extended periodsof time. Although difficult to achieve, ideal release of drugs from controlled releasedelivery systems follow zero-order release kinetics, whereby the rate of drug releasedoes not change with time until no drug remains. As a result, constant drug levelswithin the body can be maintained. A variable release rate with drugs provided to thebody at a nonconstant, time-dependent rate is more common. If first-order kineticsare followed, the release rate decreases exponentially with time until the majority ofthe drug has been released, at which time zero-order release kinetics are approached(Fig. 1.4) [9, 20–23].

1.2.3.2 Active Versus Passive Targeting. Inflammatory tissue and solidtumors both possess an increased vascular permeability that can be exploited forimproved drug delivery. The diseased tissue can be passively targeted by developingsystems (such as liposomes, micelles, and nanoparticles) with a hydrodynamic radiuslarge enough to prevent renal filtration, but small enough to pass through the leakyvasculature. In cancer, the change in vasculature is accompanied by a reduction inlymphatic drainage, thereby increasing the passive targeting capacity of carrier systemsthrough “enhanced permeation and retention” [24–26]. The site-specificity of drugdelivery systems can be further improved through the addition of a ligand, such as anantibody, polysaccharide, or peptide, that will actively target receptors overexpressedin the diseased region [27–30]. The concepts of active and passive targeting will arisethroughout this book.

1.3 BASIC PHARMACOKINETICS

1.3.1 Compartment Models

Compartment models are used as a simple method to describe the time course of adrug through a physiological system on administration. One and two compartment

BASIC PHARMACOKINETICS 9

models are depicted in Fig. 1.3. The simplest pharmacokinetic model is the one com-partment open model for drugs administered by intravenous (IV) bolus with first-orderelimination, that is, the rate at which the amount of drug in the body changes is pro-portional to the amount of drug remaining in the body. To apply a one compartmentopen model, the assumption must be made that the drugs are instantaneously, homoge-nously distributed between tissues on administration, thereby allowing the body to bedescribed as a unit from which drugs are cleared. While the one compartment modelfor IV bolus administration will be presented herein, more complicated models, suchas those required when drugs are not instantaneously distributed, are beyond the scopeof this text. Readers are encouraged to look at several excellent textbooks on basicpharmacokinetics for additional information [10, 11, 31]

As mentioned in brief above, elimination after IV bolus administration can bedescribed using a first-order kinetic equation when applying a one compartment model.This equation can be derived by assessing the rate of change for either drug concen-tration (Eq. 1.1) or drug amount (Eq. 1.2)

dCp

dt= −keCp (1.1)

dM

dt= −keM (1.2)

where Cp is the plasma concentration of drug, M is the mass amount of drug, and keis a first-order elimination rate constant. Although an identical analysis can be appliedto the rate of change of drug amount, all subsequent pharmacokinetic parameters willbe derived using the rate of change of drug concentration (Eq. 1.1). Thus, integrationof Eq. 1.1 gives:

Cp,t = Cp,0e−ket (1.3)

Equation 1.3 in conjunction with the area under the curve (AUC) described inSection 1.3.2, serves as a spring board from which other pharmacokinetic parametersare derived. Note that Cp is not equal to the concentration of drug in other tissues;

(a) (b)

DOSE DOSE

Figure 1.3. (a) One and (b) two compartment models can be used to describe the time course

of drugs in the body after administration.


however, changes in drug concentration within the plasma are directly proportionalto those in other tissues as a consequence of describing the body as a homogenous,single compartment.

1.3.2 Bioavailability and Area Under the Curve (AUC)

Bioavailability refers to the rate and extent to which a drug has reached the systemiccirculation for delivery to the site of action. Thus, the most common indicator ofbioavailability is Cp. From a plot of Cp versus time, the AUC provides a quantitativemeasure of how much drug stays in the body and for how long [10, 31].

For an IV bolus with first-order elimination kinetics, an exact solution for theAUC can be obtained by analytical integration [10, 31]. For example, consider theCp versus time plot shown in Fig. 1.4. As derived in Section 1.3.1, Cp at a giventime can be determined from Eq. 1.3. Using calculus, the AUC is equal to the inte-gral from t = 0 to an infinite time point. Therefore, taking the integral of Eq. 1.3gives

AUC =∫ ∞

0Cp,tdt (1.4)

AUC =∫ ∞

0Cp,0e−ke tdt = Cp,0

[e−ket

−ke

]∞

0(1.5)

AUC = Cp,0

[e−ke∞ − e−ke0

−ke

](1.6)

AUC = Cp,0

ke(1.7)

Cp,1

Cp

Cp,2

t1 t2 Time

Figure 1.4. After IV bolus administration, elimination can be described using a first-order

kinetic equation if a one compartment model is assumed.

BASIC PHARMACOKINETICS 11

Alternatively, Cp,0 if and/or ke are unknown, the AUC can be found using the trape-zoidal rule. Using Fig. 1.4, the AUC for the highlighted segment can be found with

AUC1−2 = Cp,1 + Cp,2

2(t2 − t1) (1.8)

Extrapolating the first segment to determine Cp,0, assuming the last points follow anexponential decay that defines ke, adding all possible segments together yields.

AUC = AUC0−1 + AUC1−last + AUClast−∞ (1.9)

AUC = Cp,0 + Cp,1

2t1 + Cp,1 + Cp,2

2(t2 − t1) + · · · + Cp,last

ke(1.10)

1.3.3 Elimination Rate Constant and Half-Life

The elimination rate constant, ke, introduced above can be found by converting Eq. 1.3to natural logarithmic form to give

Ln(Cp,t) = Ln(Cp,0) − ket (1.11)

Thus, ke is the slope of a plot of Ln(Cp) versus time:

ke = Ln(Cp,1) − Ln(Cp,2)

t2 − t1(1.12)

Note that the elimination rate constant includes both excretion and metabolism. Fromke, the half-life, that is, the time necessary to decrease Cp to one half of Cp,0, canbe determined. Considering Eq. 1.12 and solving for the time when Cp,2 = Cp,1/2gives

t1/2 = Ln2

ke= 0.693

ke(1.13)

Equation 1.13 shows that the half-life is independent of drug concentration. Thus,regardless of Cp,0, the half-life can be used to describe when most of the drug hasbeen eliminated from the body. For example, after five half-lives, Cp = Cp,0/32 and96.875% of the initial amount of drug in the body has been lost [10, 31].

1.3.4 Volume of Distribution

Despite the importance of this parameter in pharmacokinetics, the volume of distribu-tion, Vd, does not have any direct physiological relevance and does not correlate witha true volume. Vd can be defined as the ratio of dose, D, to the plasma concentrationat t = 0

Vd = D

Cp,0(1.14)


Likewise, Vd can be obtained by taking the ratio of the mass amount to theconcentration of drug at any given time point. If Vd is high, the drug is highly dis-tributed to tissues/organs throughout the body, rather than being confined primarily tothe plasma; while if Vd is low, the drug is not well distributed to tissue/organs andresides, for the most part, in the plasma [10, 31].

1.3.5 Clearance

Drug clearance (CL) is a proportionality constant relating the elimination rate, dM/dt ,to the plasma concentration Cp[10, 31].

CL = dM

dt· 1

Cp(1.15)

Substituting in Eq. 1.2 and noting that volume of distribution is equal to the amountof drug divided by the concentration of drug gives

CL = keVd (1.16)

Half-life is related to ke through Eq. 1.13. Thus,

CL = 0.693Vd

t1/2(1.17)

1.4 BASIC PHARMACODYNAMICS

1.4.1 Therapeutic Index and Therapeutic Window

The goal in the development of new therapeutic agents, as well as drug delivery sys-tems, is to maximize efficacy while minimizing the potential for adverse drug events.Thus, dose–response curves, will examine both therapeutic response and toxicity, asshown in Fig. 1.5. The ratio of the median toxic dose (TD50), that is, the dose thatcauses toxicity in 50% of the population, to the median effective dose (EC50), that is,the dose required to elicit a response in 50% of the population, is referred to as thetherapeutic index (TI). A drug with a high TI can be used over a wide range of doses,referred to as the therapeutic window, without adverse side effects. In contrast, a lowTI suggests a narrow therapeutic window [12, 32].

1.4.2 Ligand-Receptor Binding

Although some drugs act through chemical reactions or physical associations withmolecules within the body, a number of other drugs are used to elicit, change, or pre-vent a cellular response via ligand-receptor binding interactions. For this mechanismof action, the drug serves as an exogenous ligand that either (i) prevents interactions

MASS TRANSFER 13

Therapeuticwindow

50% Therapeutic index =

EC50 TD50

Dose (log scale)

EC50

TD50

% R

espo

nse

Figure 1.5. Typical dose–response curves looks at both efficacy and toxicity. The therapeutic

window is the dosing range that can be used to safely treat a disease.

of the receptor with an endogenous ligand (e.g., a cytokine or hormone), that is, thedrug acts as an antagonist, or (ii) elicits a physiological response equal to or greaterthan what would result from the binding of an endogenous ligand, that is, the drugacts as an agonist. Ligand (drug)–receptor interactions are governed by affinity, asindicated by the ratio of the association to dissociation rate constants. The inverse ofthe affinity, that is, the dissociation divided by association rate constant, is referredto as the dissociation constant (KD), the most frequently reported indicator of thestrength of drug–receptor interactions [33, 34]. The concept of ligand–receptor bind-ing is critical in understanding how to design a carrier system such that the therapeuticefficacy of the drug can be maintained and/or active targeting can be implemented.Section 1.5.2.2 takes a more quantitative approach toward helping readers understandthe importance of drug/drug delivery system–receptor interactions.

1.5 MASS TRANSFER

Learning the basics of mass transfer is critical to understanding how drugs travelthrough/out of polymeric matrices of carrier systems and through the surroundingtissue. Numerous examples using the principles of mass transfer are given throughoutthis text. Mass transfer describes the tendency of a component in a mixture to movefrom a region of high concentration (i.e., the source) to an area of low concentration(i.e., the sink). This transport can occur as a result of molecular mass transfer, ordiffusion, whereby movement occurs through a still medium, or convective masstransfer, whereby transfer is promoted by fluid flow. The interested reader is referredto conventional texts on mass transfer and transport phenomena [35–37].

1.5.1 General Flux Equation and Fick’s First Law

The total mass transported can be expressed as the sum of the mass transported bydiffusion and the mass transported by bulk motion of the fluid. Considering a mixture


of two species with one dimensional transport along the z axis, the molar flux ofspecies 1, N1, is given by

N1 = J1 + c1v∗ (1.18)

where J1 is the flux due to pure diffusion, c1 is the molar concentration, and v∗ is themolar average velocity. v∗ can be determined as the sum of the velocity contributionsfrom the components in the mixture.

v∗ = 1

c

∑icivi =

∑ixivi (1.19)

where xi is the mole fraction of species i in the mixture. civi is equivalent to themolar flux of species i relative to stationary coordinates.

Ni = civi

(mol

m2s

)(1.20)

Thus, in a binary mixture

v∗ = 1

c

∑icivi = 1

c(c1v1 + c2v2) = 1

c(N1 + N2) (1.21)

Referring back to Eq. 1.18, c1v∗ is the flux generated by processes other than diffusion,

such as convection/fluid flow. The flux, owing to diffusion, J1, can also be expressedin the form of Fick’s First Law in one dimension

J1 = −D1dc1

dz(1.22)

where D1 is a proportionality constant referred to as the diffusion coefficient. Com-bining Eqs. 1.18, 1.21, and 1.22 yields the General Flux Equation [35, 36]:

N1 = −D1dc1

dz+ c1

c(N1 + N2) (1.23)

Of note, for dilute solutions, as would be found for a drug moving though a polymermatix or tissue, the general flux equation reduces to Fick’s first law.

N1 = −D1dc1

dz(1.24)

MASS TRANSFER 15

1.5.2 Mass Conservation and Fick’s Second Law

Referring to Fig. 1.6, consider a material balance on species 1 along diffusion pathlength z and through fixed cross sectional area for flux A.

By conservation of mass in − out + generation = accumulation, expressed math-ematically as

N1A|z − N1A|z+�z + ψ1A�Z = c1|t+�t,z − c1|t,z�t

A�Z (1.25)

Division of Eq. 1.25 by A, rearrangement, and division by �Z yields

−[N1|z+�z − N1|z

�Z

]+ ψ1 = c1|t+�t,z − c1|t,z

�t(1.26)

If the limit of �Z→ 0, �t → 0 is taken, the following equation is obtained:

−∂N1

∂z+ ψ1 = ∂c1

∂t(1.27)

Using Eq. 1.27 with the General Flux Equation (Eq. 1.23), assuming that D1 isconstant, gives

D1∂2c1

∂z2− ∂(c1v

∗)∂z

+ ψ1 = ∂c1

∂t(1.28)

If the total system density is also constant, Eq. 1.28 can be further simplified to

D1∂2c1

∂z2− v∗ ∂c1

∂z+ ψ1 = ∂c1

∂t(1.29)

In a situation with no fluid motion (v∗ = 0) and no productive term (ψ1 = 0), thisequation reduces to Fick’s Second Law, which facilitates prediction of concentrationchanges with time because of diffusion [35, 36].

D1∂2c1

∂z2= ∂c1

∂t(1.30)

In N1 Out N1

Surface A

Z Z+ΔZ

Figure 1.6. A generalized mass balance for a volume element.


Rectangular Cylindrical Spherical

y

z

x

zrθ

θ

ϕ

y

x

z

r

y(a) (b) (c)

Figure 1.7. One dimensional flux equations can be derived for (a) rectangular, (b) cylindrical,

and (c) spherical geometries.

TABLE 1.3. One Dimensional Flux Equations for DifferentGeometries

Rectangular D1∂2c1

dz 2 = ∂c1

∂t

Cylindrical D1

[∂2c1

dr2 + 1

r

∂c1

∂r

]= ∂c1

∂t

Spherical D1

[∂2c1

dr2 + 2

r

∂c

∂r

]= ∂c1

∂t

Although Fick’s Second Law was derived for one dimension flux in a rectangularcoordinate system above, these concepts can readily be extended to spherical andcylindrical coordinate systems (Fig. 1.7). The equations for one dimensional flux indifferent geometries are summarized in Table 1.3. Detailed derivations of solutions toFick’s Second Law, including those given for the problems in Section 1.5.2.1, can befound in Crank’s book on the mathematics of diffusion [38].

1.5.2.1 Application of Fick’s Second Law in Drug Delivery. Applicationsof Fick’s Second Law will appear throughout this text; however, two in depth exampleswill be provided to here to show how Eq. 1.30 can be used to predict the concentrationof drug as a function of time and distance away from or through a controlled releasesystem. First, consider a cylindrical hydrogel with a radius of 4 mm and a heightof 0.75 mm loaded with keratinocyte growth factor (KGF) at a high concentration(cKGF,0) intended for use as a wound healing dressing (Fig. 1.8) [39]. Assuming thatdiffusion only occurs in one dimension through the surface placed in contact with thewound and taking into account that h � r , the system can be modeled with Fick’sSecond Law using a rectangular coordinate system.

D1∂2c1

dz2= ∂c1

∂t(1.31)

MASS TRANSFER 17

h

NKGF

rcKGF,0

z

x

z = 0y

Figure 1.8. KGF release from a cylindrical hydrogel with h << r can be modeled as one

dimensional flux in the z direction (i.e., a rectangular coordinate system).

If we assume that (i) a high concentration of drug is maintained at the surface of thecylinder, (ii) KGF is not initially present in the underlying tissue, and (iii) there is noKGF at an infinite distance from the cylinder, the following boundary conditions canbe applied to determine the drug concentration as a function of time and distance intothe underlying tissue.

cKGF(z, 0) = 0 for 0 < z < ∞, t = 0 (1.32)

cKGF(0, t)(surface) = cKGF,0 for z = 0, t > 0 (1.33)

cKGF(∞, t) = 0 for z = ∞, t > 0 (1.34)

Solving Eq. 1.31 with the method of combination of variables gives the followingsolution

cKGF(z, t) − cKGF(z, 0)

cKGF,0 − cKGF(z, 0)= Erfc

(z

2√

DKGFt

)(1.35)

Which, given that cKGF(z, 0) = 0, can be reduced to

cKGF(z, t)

cKGF,0= Erfc

(z

2√

DKGFt

)(1.36)

Taking DKGF to be 4.86×10−9 cm2 s−1, the concentration of KGF as a function ofdistance from the hydrogel wound healing dressing is plotted for several time pointsin Fig. 1.9.

Next, consider the release of 10 mg of Dramamine from a spherical capsule (r =0.30 cm) (Fig. 1.10). Using a spherical coordinate system and assuming that diffusiononly occurs in the radial direction, Fick’s Second Law can be used to predict thechange in drug concentration within the capsule over time.

D1

[∂2c1

dr2+ 2

r

∂c

∂r

]= ∂c1

∂t(1.37)


1.2

1

0.8

24 h12 h6 h

0.6

0.4c

KG

F (

z,t)

/cK

GF

,00.2

00 0.02 0.04 0.06

Distance (cm)

0.08 0.1

Figure 1.9. Distance of penetration of KGF into the wound site following release from

cylindrical hydrogels at three time points (6, 12, and 24 h), as determined from Eq. 1.36.

cdramamine,0

Ndramamine

r = R

r

Figure 1.10. Release of Dramamine from a spherical capsule can be model as one dimensional

flux in the radial direction.

The following boundary conditions can be applied assuming that (i) the capsule radiusremains constant, (ii) the capsule possesses radial symmetry, and (iii) the drug isimmediately swept away from the surface of the capsule on release.

cd(r, 0) = cd,0 for 0 < r < R, t = 0 (1.38)

∂cd(0, t)

∂r= 0 for r = 0, t ≥ 0 (1.39)

cd(R, t)(surface) = 0 for r = R, t > 0 (1.40)

An analytical solution to Eq. 1.37 can be obtained following the separation of variablesmethod.

cd(r, t) − cd,0

cd(R, t) − cd,0= 1 + 2R

πr

∞∑n=1

−1n

nsin

(nπr

R

)e

−Ddn2π2 t

R2 (1.41)

MASS TRANSFER 19

100

8080

60

40

cd

(mg

ml−1

)20

00 0.05 0.1 0.15 0.2

Radius (cm)

2 h6 h12 h24 h

0.25 0.3 0.35

Figure 1.11. Dramamine concentration throughout the spherical capsule was predicted for

several time points based on Eq. 1.42.

which, given that cdramamine(R, t)= 0, can be simplified to

cd(r, t) − cd,0

−cd,0= 1 + 2R

πr

∞∑n=1

−1n

nsin

(nπr

R

)e

−Ddn2π2 t

R2 (1.42)

Figure 1.11 illustrates the change in Dramamine concentration with distance out-ward from the center of the capsule for several different time points. Alternatively,by using r = R and md,0 = cd,0 × (4/3)πR3, where md,0 is the initial mass amountof Dramamine loaded into the capsule, the equation can be revised to predict the timenecessary for near complete drug release. Figure 1.12 uses Eq. 1.43 to demonstratethe fractional release of drug (1−md(t)/md,0) as a function of time.

md(t)

md,0= 6

π2

∞∑n=1

1

n2e

−Ddn2π2 t

R2 (1.43)

1.5.2.2 Fick’s Second Law and Ligand Binding. As discussed previously,there are many instances, particularly in regard to biologics; efficacy is dependent onthe therapeutic agent not only diffusing to the cells within the active site, but also onbinding to the cell surface. For these cases, the assumption cannot be made that thedrug disappears immediately on reaching the cell, that is, the drug concentration atthe surface is equal to 0. Instead, the drug disappears at a rate that is governed bybinding kinetics.

Consider the system illustrated in Fig. 1.13. At the surface, the drug can bind toor dissociate from the receptor. This relationship can be described by

CR + CLkon�koff

CRCL (1.44)


1

0.8

0.61−

md/

md,

0

0.4

0.2

00 5 10 15

Time (h)

20 25 30 35

Figure 1.12. Equation 1.43 can be used to predict when most of the Dramamine will be

released from the spherical capsule.

k+k−

konkoff

kf

kr

Figure 1.13. For a ligand (i.e., drug) to associate with a cell surface receptor, the drug must

first diffuse to the cell surface.

where kon and koff are the rate constants of binding and dissociation, respectively;CR is the concentration of the receptor; CL is the concentration of ligand (drug); andCRCL is the concentration of ligand bound to receptor. At equilibrium, CR, CL, andCRCL remain unchanged with time. Thus,

d(CRCL)

dt= CR · CL · kon − CRCL · koff = 0 (1.45)

CR · CL · kon = CRCL · koff (1.46)

Equation 1.46 can be rearranged and expressed with the equilibrium dissociationconstant, Kd.

CR·CL

CRCL= koff

kon= Kd (1.47)

MASS TRANSFER 21

Likewise, an equilibrium exists between the drug diffusing to and from the recep-tor, as defined by k+ and k−. Taken together, the overall forward and reverse rateconstants are given by kf and kr, respectively.

By assuming that (i) flux only occurs in the radial direction, (ii) the ligand doesnot degrade within the physiological solution, (iii) the cell radius remains constant,and (iv) the rate of ligand disappearance is equal to the rate of diffusion at the surface,expressions can be developed to determine kf and kr. Because there is a constant source(the ligand in solution) and a constant sink (the cell surface), the system is at steadystate. Thus, Fick’s second law for a spherical geometry can be written as

DL

[d2cL

dr2+ 2

r

dcL

dr

]= 0 (1.48)

The following boundary conditions can be applied based on the assumptions givenabove.

cL(r) = cL,0 for r = ∞ (1.49)

4πR2 · NL = kon · CR · CL for r = R (1.50)

The second boundary condition equates the rate of ligand disappearance at the surface,as given by kon × CR × CL, to the rate of diffusion at the surface, as given by thesurface area (4πR2) times the flux (NL). Thus, this boundary condition can be rewrittenas

4πR2 · DLdcL

dr= kon · CR · CL for r = R (1.51)

Solving with the specified boundary conditions yields the ligand concentration as afunction of radius.

CL(r) = −konCRRCL,0

4πDr + konCR· 1

r+ CL,0 (1.52)

If binding is diffusion-limited, that is, 4πDr � kon, the rate of ligand disappearanceat the cell surface (r = R) can be given by

Rate of ligand disappearance = −D(4πR 2)dCL

dr(1.53)

Substituting in Eq. 1.52 into Eq. 1.53 gives

Rate of ligand disappearance = 4πDkonCRRCL,0

4πDR + konCR(1.54)

where 4πDR is equivalent to k+. By equating the overall rate of ligands diffusingtoward and binding to the cell surface receptors, kf CL,0, to Eq. 1.54, the overall rate


constant kf can be expressed in terms of k+, the rate constant for diffusion-limitedbinding, to CRkon, the intrinsic binding rate [40, 41].

kf = k+CRkon

k+ + konCR(1.55)

As an example, consider that antibody fragments conjugated to PEG can beused for the active, targeted delivery of therapeutics to cancer cells that possessspecific cell surface antigens. The hydrodynamic radius and diffusion coefficient ofthe antibody-PEG fragment in PBS at 37 ◦C have been determined to be 2.5 nmand 8.4×10−7 cm2 s−1 respectively. The intrinsic association constant, kon, is6.1×104 M−1 s−1 [42]. Assuming that binding is diffusion limited, the transportrate constant, k+, and the overall rate constants for ligand binding, kf, for a normalcell that has 20,000 surface receptors (CR = 20,000) and a cancerous cell that has2,000,000 receptors (CR = 2,000,000) can be determined.

k+ = 4πDR

k+ = 4π(8.4 × 10−7 cm2 s−1)(2.5 × 10−7 cm)

k+ = 2.64 × 10−12 cm3 s−1 ligand−1

k+ = 2.64 × 10−12 cm3

ligand× 6.022 × 1023 ligands

mole× 1l

1000 cm3

k+ = 1.59 × 109 M−1 s−1

kf = k+CRkon

k+ + CRkon

For normal cells:

CR = 20, 000

kf = 6.90 × 108 M−1 s−1

For cancer cells:

CR = 2, 000, 000

kf = 1.57 × 109 M−1 s−1

Thus, the overall forward rate constant for cancer cells is greater than that for normalcells, lending credence to the possibility of active targeting by carrier systems modifiedwith a ligand for receptors overexpressed by diseased cells. Note that while the abovecalculations were made on a per cell basis, careful attention should be given to unitswhen solving for problems related to ligand binding interactions.

HOMEWORK PROBLEMS 23

1.6 KEY POINTS

• As drug discovery has evolved to encompass compounds that are less physio-logically soluble and/or stable, the need for drug delivery has increased.

• Pharmacokinetics refers to what the body does to the drug, while pharmacody-namics refers to what the drug does to the body.

• The route of absorption can have a profound impact on pharmacokinetic prop-erties.

• Compartmental models can be used to describe the absorption, distribution,metabolism, and excretion of drugs by/from the body.

• In many cases, Fick’s second law can be used to predict the release of a drugfrom a polymer-based drug delivery system.

1.7 HOMEWORK PROBLEMS

1. Discuss why 100% bioavailability is difficult to obtain by oral drug delivery.

2. Apo2L/TRAIL (tumor necrosis factor-related apoptosis-inducing ligand) hasdemonstrated anticancer efficacy. Recently, a recombinant, water soluble formof Apo2L/TRAIL was developed for clinical application. Before clinical stud-ies, several in vivo models were used for pharmacokinetic evaluation. Forall animals, Apo2L/TRAIL was administered via an IV bolus. The followingaverage data was obtained from chimpanzees administered Apo2L/TRAIL ata dose of 1 mg−1 kg [43].

Time, min Cp, ng ml−1

10 20,00020 15,00045 900060 600090 2000

120 900180 200

Construct a semi-log plot of serum concentration versus time and determinethe best fit exponential equation for the curve. Determine the following phar-macokinetic parameters, assuming a chimpanzee weight of 60 kg:a. Elimination rate constant, ke

b. Half-life, t1/2

c. Volume of distribution, Vd

d. Clearance, CL


3. The diffusion coefficients for the antibiotic cefoperazone through agar gel,fibrin gel, and cerebral cortex tissue were determined by applying a solution ofdrug in PBS at a concentration of 5 mg ml−1 to the top of the appropriate matrixand measuring the concentration as a function of depth at a predetermined timepoint. Experiments with brain tissue were performed in vivo on rats, whileexperiments with agar and fibrin gel were performed on matrix prepared inPetri dishes (thickness = 0.5 cm, diameter = 10 cm). The following data wasobtained [44]:

Matrix D, cm2 s−1

Agar gel 6.10E−07Fibrin gel 7.00E−07Cortex tissue 2.50E−08

a. Construct a model of cefoperazone penetration into agar, fibrin, or cor-tex tissue by (i) drawing the physical situation, (ii) listing at least threeassumptions, (iii) specifying the boundary and initial conditions, and (iv)formulating the correct differential equation for mass transfer.

b. Assuming that the correct differential equation and boundary/initial condi-tions were identified, the analytic solution is

cCefazolin(z, t) = cCefazolin,0 × Erfc

(z

2√

Dt

)

Construct plots showing (i) the concentration of cefoperazone as a functionof depth (0–500 μm) at a time of 30 min and (ii) the concentration ofcefoperazone as a function of time (5–30 min) at a depth of 100 μm foragar gel, fibrin gel, and cortex tissue.

4. To control inflammation around implantable glucose sensors, researchers havesuggested controlled release of dexamethasone at the site of implantation.In an experimental study with rats, dexamethasone was released fromosmotic pumps implanted subcutaneously. Drug delivery from the pump wasachieved from the spherical tip (radius = 0.6 mm) of a catheter attachedto the pump. The osmotic pump maintains a constant concentration in thecatheter tip. The following data was obtained for the concentration versusdistance profile of dexamethasone at 6 h after implantation. Distance isexpressed as the radial distance (r) from the center of the catheter tip, whileconcentration in the tissue is expressed relative to the concentration at the tip(Cs) [45].

HOMEWORK PROBLEMS 25

r (distance from the centerof the catheter tip, cm)

C/Cs

0.06 10.085 0.720.110 0.540.135 0.320.16 0.250.21 0.130.26 0.110.31 0.0730.36 0.060.41 0.05

a. Construct a model to describe the controlled release of dexamethasone fromthe spherical tip of the catheter. Draw a picture of the physical system,list at least three assumptions, decide on the most appropriate coordinatesystem, formulate the differential equation for mass transfer, and specifythe boundary/initial conditions.

b. The solution for the release of dexamethasone from the spherical tip of thecatheter can be given by

C

Cs= a

rerfc

[r − a

2√

Dt

]

where a is the radius of catheter, r is the distance from the center ofthe catheter, and C/Cs is the concentration in the tissue relative to theconcentration at the tip.Given the information about the radius of the catheter tip and the concen-tration profile of dexamethasone obtained at a time point of 6 h, plot theabove equation and determine the diffusion coefficient for dexamethasone.

c. Evidence suggests that dexamethasone can be eliminated from the site ofimplantation. To account for this, an elimination rate constant (ke) can beincorporated into the solution for predicting the concentration as a functionof radius.

C

Cs= a

2r

{exp

[− (r − a)

√k

D

]erfc

[r − a

2√

Dt−

√kt

]

+ exp

[(r − a)

√k

D

]erfc

[r − a

2√

Dt+

√kt

]}


Plot this equation using the diffusion coefficient determined in part (b).Comment on whether or not you think elimination plays a significant rolein the observed concentration profile.

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